The reports about siwak and purification share a concern for cleanliness, but they do not cover exactly the same scope. Siwak is a specific practice of cleaning the mouth; purification includes the wider requirements of ritual cleanliness.
Example 1: Cleaning the teeth is oral hygiene; it does not by itself replace the requirements of wudu or ghusl.
Example 2: A person who prepares for prayer through proper purification can also maintain daily oral hygiene.
Recognising this difference allows مَطْهَرَةٌ لِلْفَمِ and الطُّهُورُ شَطْرُ الْإِيمَانِ to be understood in their distinct contexts.